Characters Of Minimal Models Research Articles

N-th parafermion {mathcal{W}}_N characters from U(N) instanton counting on \u21022/\u2124n

We propose, following the AGT correspondence, how the {mathcal{W}}_{N,n}^{mathrm{para}} (n-th parafermion {mathcal{W}}_N ) minimal model characters are obtained from the U(N ) instanton counting on ℂ2/ℤn with Ω-deformation by imposing specific conditions which remove the minimal model null states.

Closed form fermionic expressions for the Macdonald index

We interpret aspects of the Schur indices, that were identified with characters of highest weight modules in Virasoro (p, p′) = (2, 2k + 3) minimal models for k = 1, 2, . . . , in terms of paths that first appeared in exact solutions in statistical mechanics. From that, we propose closed-form fermionic sum expressions, that is, q, t-series with manifestly non-negative coefficients, for two infinite-series of Macdonald indices of (A1, A2k ) Argyres- Douglas theories that correspond to t-refinements of Virasoro (p, p′) = (2, 2k + 3) minimal model characters, and two rank-2 Macdonald indices that correspond to t-refinements of {mathcal{W}}_3 non-unitary minimal model characters. Our proposals match with computations from 4d mathcal{N} = 2 gauge theories via the TQFT picture, based on the work of J Song [75].

Cylindric partitions, characters and the Andrews–Gordon–Bressoud identities

We study the Andrews–Gordon–Bressoud (AGB) generalisations of the Rogers–Ramanujan q-series identities in the context of cylindric partitions. We recall the definition of r-cylindric partitions, and provide a simple proof of Borodin’s product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler. We also recall the relationships between the r-cylindric partition generating functions, the principal characters of algebras, the minimal model characters of algebras, and the r-string abaci generating functions, providing simple proofs for each. We then set r = 2, and use two-cylindric partitions to re-derive the AGB identities as follows. Firstly, we use Borodin’s product expression for the generating functions of the two-cylindric partitions with infinitely long parts, to obtain the product sides of the AGB identities, times a factor , which is the generating function of ordinary partitions. Next, we obtain a bijection from the two-cylindric partitions, via two-string abaci, into decorated versions of Bressoud’s restricted lattice paths. Extending Bressoud’s method of transforming between restricted paths that obey different restrictions, we obtain sum expressions with manifestly non-negative coefficients for the generating functions of the two-cylindric partitions which contains a factor . Equating the product and sum expressions of the same two-cylindric partitions, and canceling a factor of on each side, we obtain the AGB identities.

Half-lattice Paths and Virasoro Characters

We first briefly review the role of lattice paths in the derivation of fermionic expressions for the M(p,p') minimal model characters of the Virasoro Lie algebra. We then focus on the recently introduced half-lattice paths for the M(p,2p+/-1) characters, reformulating them in such a way that the two cases may be treated uniformly. That the generating functions of these half-lattice paths are indeed M(p,2p+/-1) characters is proved by describing weight preserving bijections between them and the corresponding RSOS lattice paths. Here, the M(p,2p-1) case is derived for the first time. We then apply the methods of Bressoud and Warnaar to these half-lattice paths to derive fermionic expressions for the Virasoro characters X^{p,2p+/-1}_{1,2} that differ from those obtained from the RSOS paths. This work is an extension of that presented by the third author at the "7th International Conference on Lattice Path Combinatorics and Applications", Siena, Italy, July 2010.

FACTORIZED CHARACTERS AND FORM FACTORS OF DESCENDANT OPERATORS IN PERTURBED CONFORMAL SYSTEMS

We determine which characters of minimal models factorize into an infinite product like the one of the spin operator in the Ising model. In particular, all characters in the Ising model and in the nonunitary minimal Lee-Yang model factorize. This information is used to determine form factors of descendant operators, especially in the scaling Lee-Yang model.